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A213158
Positive integers of the form (x+y+z)*x*y*z (x,y,z positive integers).
2
3, 8, 15, 20, 24, 35, 36, 48, 56, 63, 80, 84, 96, 99, 108, 120, 128, 135, 140, 143, 144, 168, 176, 180, 195, 200, 216, 224, 231, 240, 243, 255, 260, 264, 275, 288, 300, 308, 320, 323, 336, 351, 360, 384, 396, 399, 416, 420, 440, 455, 468, 476, 483, 495, 504
OFFSET
1,1
COMMENTS
Square terms are 36, 144, 576,... and the corresponding square roots are 6, 12, 24,... i.e. sequence A188158 (integer areas of primitive integer triangles).
Positive integers of the form (a^2-b^2)*(b^2-c^2) with integers a>b>c>=0. - Michael Somos, May 18 2013
REFERENCES
R. D. Carmichael, Diophantine Analysis, Wiley, 1915, p. 9.
EXAMPLE
a(21)=144 for x=1, y=4 and z=4 then the triangle sides are x+y = 5, z+x = 5 and y+z = 8, hence half-perimeter = p = x+y+z = 9 and Heron's formula is checked: area = sqrt(p*(p-5)*(p-5)*(p-8)) = sqrt(144) = 12.
36 = (4^2-2^2) * (2^2-1^2). 63 = (5^2-2^2) * (2^2-1^2) = (5^2-4^2) * (4^2-2^2)= (8^2-1^2) * (1^2-0^2). - Michael Somos, May 19 2013
MATHEMATICA
nmax = 25; mx = nmax (nmax + 2); Union[Reap[Do[a = (x + y + z)*x*y*z; If[a <= mx, Sow[a]], {x, 1, nmax}, {y, x, nmax}, {z, y, nmax}]][[2, 1]]]
CROSSREFS
Cf. A188158.
Sequence in context: A310311 A310312 A310313 * A228880 A310314 A181027
KEYWORD
nonn
AUTHOR
STATUS
approved